These complex number algorithms are the inverses of trigonometric functions
currently present in the C++ standard. Equivalents to these functions
are part of the C99 standard, and are part of the Technical
Report on C++ Library Extensions.

The class and function templates in <boost/math/common_factor.hpp>
provide run-time and compile-time evaluation of the greatest common
divisor (GCD) or least common multiple (LCM) of two integers. These
facilities are useful for many numeric-oriented generic programming
problems.

In practical terms, an octonion is simply an octuple of real numbers
(α,β,γ,δ,ε,ζ,η,θ), which we can write in the form o = α + βi + γj + δk + εe' + ζi' + ηj' + θk',
where i, j
and k are the same objects
as for quaternions, and e',
i', j'
and k' are distinct objects
which play essentially the same kind of role as i
(or j or k).

Addition and a multiplication is defined on the set of octonions, which
generalize their quaternionic counterparts. The main novelty this time
is that the multiplication is not only not commutative,
is now not even associative (i.e. there are quaternions
x, y
and z such that x(yz)
≠ (xy)z). A way of remembering things is by using
the following multiplication table:

Some traditional constructs, such as the exponential, carry over without
too much change into the realms of octonions, but other, such as taking
a square root, do not (the fact that the exponential has a closed form
is a result of the author, but the fact that the exponential exists
at all for octonions is known since quite a long time ago).

Provides a number of high quality special functions, initially these
were concentrated on functions used in statistical applications along
with those in the Technical Report on C++ Library Extensions.

The function families currently implemented are the gamma, beta &
erf functions along with the incomplete gamma and beta functions (four
variants of each) and all the possible inverses of these, plus digamma,
various factorial functions, Bessel functions, elliptic integrals,
sinus cardinals (along with their hyperbolic variants), inverse hyperbolic
functions, Legrendre/Laguerre/Hermite polynomials and various special
power and logarithmic functions.

All the implementations are fully generic and support the use of arbitrary
"real-number" types, although they are optimised for use
with types with known-about significand (or mantissa) sizes: typically
float, double or long double.

Quaternions are in fact part of a small hierarchy of structures built
upon the real numbers, which comprise only the set of real numbers
(traditionally named R),
the set of complex numbers (traditionally named C), the set of quaternions (traditionally
named H) and
the set of octonions (traditionally named O),
which possess interesting mathematical properties (chief among which
is the fact that they are division algebras,
i.e. where the following property is true: if
y is an element of that algebra
and is not equal to zero, then yx
= yx', where x
and x' denote elements of that
algebra, implies that x = x').
Each member of the hierarchy is a super-set of the former.

One of the most important aspects of quaternions is that they provide
an efficient way to parameterize rotations in R3
(the usual three-dimensional space) and R4.

In practical terms, a quaternion is simply a quadruple of real numbers
(α,β,γ,δ), which we can write in the form q = α + βi + γj + δk,
where i is the same object
as for complex numbers, and j
and k are distinct objects
which play essentially the same kind of role as i.

An addition and a multiplication is defined on the set of quaternions,
which generalize their real and complex counterparts. The main novelty
here is that the multiplication is not commutative
(i.e. there are quaternions x
and y such that xy
≠ yx). A good mnemotechnical way of remembering
things is by using the formula i*i = j*j = k*k =
-1.

As implied by its name, this library is intended to help manipulating
mathematical intervals. It consists of a single header <boost/numeric/interval.hpp>
and principally a type which can be used as interval<T>.

The header <boost/operators.hpp> supplies several sets of class
templates (in namespace boost). These templates define operators at
namespace scope in terms of a minimal number of fundamental operators
provided by the class.

Random numbers are useful in a variety of applications. The Boost Random
Number Library (Boost.Random for short) provides a vast variety of
generators and distributions to produce random numbers having useful
properties, such as uniform distribution.